C₅×#6514 #11765109

order	1120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#6514

Splittings

Hall	C₅×C₇ ⋊ #137
Sylow	C₇ ⋊ C₅×#137

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂×C₂²⋊D₇
commutator, cosocle	C₄×C₇	C₂³×C₅
Frattini	C₄	C₂²×C₅×D₇
Fitting, prime radical	C₅×C₇×#41	C₂
socle	C₂×C₅×C₇	C₂×D₄
hypercenter	C₅×#41	D₇
hypocenter	C₇	C₅×#137
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#41

120 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 560

A minimal faithful permutation representation — 3 generators on 28 points:

(1,2,3,4,5)(6,7,9,11,12,10,8)(14,19)(15,22)(16,24)(18,25)(20,21)(26,27);(13,15)(14,18)(16,23)(17,22)(19,27)(20,25)(21,26)(24,28);(7,8)(9,10)(11,12)(13,14,17,21)(15,18,22,26)(16,19,24,20)(23,27,28,25)

Conjugacy classes & character table

205 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 715);